91Ó°ÊÓ

In Problems \(61-64\), use a graphing utility to plot the first ten terms of the given sequence. $$ \left\\{\frac{4^{n}}{n !}\right\\} $$

Use the Binomial Theorem to show that $$ \left(\begin{array}{l} n \\ 0 \end{array}\right)+\left(\begin{array}{l} n \\ 1 \end{array}\right)+\cdots+\left(\begin{array}{l} n \\ n \end{array}\right)=2^{n} $$

Prove that \(\left(\begin{array}{c}n \\ r-1\end{array}\right)+\left(\begin{array}{l}n \\\ r\end{array}\right)=\left(\begin{array}{c}n+1 \\ r\end{array}\right), \quad 0

In Problems \(61-64\), use a graphing utility to plot the first ten terms of the given sequence. $$ \left\\{(-1)^{n} \frac{5}{n}\right\\} $$

In Problems \(61-64\), use a graphing utility to plot the first ten terms of the given sequence. $$ \left\\{(-1)^{n-1} \frac{10 n}{n+3}\right\\} $$

Prove that \(\left(\begin{array}{c}n \\\ r+1\end{array}\right)=\frac{n-r}{r+1}\left(\begin{array}{l}n \\\ r\end{array}\right), \quad 0 \leq r

Find two different values of \(x\) such that \(-\frac{3}{2}, x,-\frac{8}{27}, \ldots\) is a geometric sequence.

If \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) are geometric sequences, then show that \(\left\\{a_{n} b_{n}\right\\}\) is a geometric sequence.

Find a piecewise-defined formula for the general term \(a_{n}\) of the sequence $$ 1, \frac{1}{3}, \frac{1}{3}, \frac{1}{5}, \frac{1}{5}, \frac{1}{7}, \frac{1}{7}, \ldots $$

In Problems 69 and 70 , find a value of \(a_{1}\) and a recursion formula that defines the given sequence. Find \(a_{5}\) of each sequence. $$ \sqrt{3}, \sqrt{3+\sqrt{3}}, \sqrt{3+\sqrt{3+\sqrt{3}}}, \sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}, \ldots $$